Chapter 1. Pigeonhole Principle Prof. Tesler Math 184A Winter 2019 Prof. Tesler Ch. 1. Pigeonhole Principle Math 184A / Winter 2019 1 / 19
Pigeonhole principle https://commons.wikimedia.org/wiki/File:TooManyPigeons.jpg If you put 10 pigeons into 9 holes, at least one hole has at least two pigeons. Pigeonhole Principle If you put n items into k boxes, for integers n > k > 0 , then at least one box has 2 or more items. Prof. Tesler Ch. 1. Pigeonhole Principle Math 184A / Winter 2019 2 / 19
Example: Five real numbers in [ 0 , 1 ] x 1 , . . . , x 5 ∈ [ 0 , 1 ] . Pick five real numbers between 0 and 1 : We will show that at least two of them differ by at most 1 / 4 . Split [ 0 , 1 ] into 4 bins of width 1 / 4 : 0 1/4 1/2 3/4 1 Put 5 numbers into 4 bins. At least two numbers are in the same bin, and thus, are within 1 / 4 of each other. Technicality: We didn’t say how to handle 1 4 , 1 2 , and 3 4 . 4 in the 1 st or 2 nd bin? However, it doesn’t affect the conclusion. Is 1 Generalization For an integer n � 2 , pick n numbers between 0 and 1 . 1 Then at least two of them are within n − 1 of each other. Prof. Tesler Ch. 1. Pigeonhole Principle Math 184A / Winter 2019 3 / 19
Example: Picking 6 numbers from 1 to 10 Pick 6 different integers from 1 to 10. Claim: Two of them must add up to 11. Example 1, 4, 7, 8, 9, 10 has 1 + 10 = 11 and 4 + 7 = 11 . Proof. Pair up the numbers from 1 to 10 as follows: ( 1 , 10 ) ( 2 , 9 ) ( 3 , 8 ) ( 4 , 7 ) ( 5 , 6 ) There are 5 pairs, each summing to 11. At least two of the 6 numbers must be in the same pair. � Prof. Tesler Ch. 1. Pigeonhole Principle Math 184A / Winter 2019 4 / 19
Example: 5 points on a sphere Select any five points on a sphere. Then the sphere may be cut in half (both halves include the boundary) so that one of the two hemispheres has at least four of the points. Proof: Pick any two of the five points. Form the great circle through those points, splitting it into two hemispheres. Three of the five points remain. One of the hemispheres contains at least two of these points, plus the original two points. Thus, it contains at least four of the points. Prof. Tesler Ch. 1. Pigeonhole Principle Math 184A / Winter 2019 5 / 19
Terminology In the last two examples: “pigeons” are numbers 1 , . . . , 10 , and “holes” are ordered pairs; “pigeons” are points on a sphere and the “holes” are hemispheres. “Pigeons” and “holes” are abstractions, and may be replaced by other terms: Place pigeons into holes. Place balls into boxes. Place items into bins. In any particular problem, be sure to Properly identify what plays the roles of “pigeons” and “holes.” Give the rule for placing pigeons into holes. Prof. Tesler Ch. 1. Pigeonhole Principle Math 184A / Winter 2019 6 / 19
Notation Let x be a real number. ⌊ x ⌋ = floor of x = largest integer that’s � x ⌈ x ⌉ = ceiling of x = smallest integer that’s � x ⌊ 2 . 5 ⌋ = 2 ⌊ − 2 . 5 ⌋ = − 3 ⌊ 2 ⌋ = ⌈ 2 ⌉ = 2 ⌈ 2 . 5 ⌉ = 3 ⌈ − 2 . 5 ⌉ = − 2 ⌊ − 2 ⌋ = ⌈ − 2 ⌉ = − 2 You have probably seen [ x ] used for greatest integer , and you may encounter some differences in definitions for negative numbers. However, we will use the floor and ceiling notation above, and will use square brackets for [ n ] = { 1 , 2 , 3 , . . . , n } (for integers n � 1 ) [ 0 ] = ∅ . and This notation is common in Combinatorics. Prof. Tesler Ch. 1. Pigeonhole Principle Math 184A / Winter 2019 7 / 19
Generalized Pigeonhole Principle Put n items into k boxes (integers n � 0 and k � 1 ). Then There is a box with at least ⌈ n / k ⌉ items. There is a box with at most ⌊ n / k ⌋ items. Example of Generalized Pigeonhole Principle 150 pigeons are placed into 60 holes. At least one hole has ⌈ 150 / 60 ⌉ = ⌈ 2 . 5 ⌉ = 3 or more pigeons. At least one hole has ⌊ 150 / 60 ⌋ = ⌊ 2 . 5 ⌋ = 2 or fewer pigeons. Relation to basic version of the pigeonhole principle The basic version of the pigeonhole principle is for n > k > 0 . Then ⌈ n / k ⌉ � n / k > 1 , so ⌈ n / k ⌉ � 2 . So there is a box with at least 2 items. Prof. Tesler Ch. 1. Pigeonhole Principle Math 184A / Winter 2019 8 / 19
Proof for this example Example 150 pigeons are placed into 60 holes. At least one hole has ⌈ 150 / 60 ⌉ = ⌈ 2 . 5 ⌉ = 3 or more pigeons. At least one hole has ⌊ 150 / 60 ⌋ = ⌊ 2 . 5 ⌋ = 2 or fewer pigeons. Proof that at least one hole has 3 or more. Suppose all holes have at most 2 pigeons. Then combined, the holes have at most 2 ( 60 ) = 120 pigeons. But they have 150 pigeons, and 120 < 150 , a contradiction. Thus, some hole has at least 3 pigeons. � Prof. Tesler Ch. 1. Pigeonhole Principle Math 184A / Winter 2019 9 / 19
Proof for this example Example 150 pigeons are placed into 60 holes. At least one hole has ⌈ 150 / 60 ⌉ = ⌈ 2 . 5 ⌉ = 3 or more pigeons. At least one hole has ⌊ 150 / 60 ⌋ = ⌊ 2 . 5 ⌋ = 2 or fewer pigeons. Proof that at least one hole has 2 or fewer. Suppose all holes have at least 3 pigeons. Then combined, the holes have at least 3 ( 60 ) = 180 pigeons. But they have 150 pigeons, and 180 > 150 , a contradiction. Thus, some hole has at most 2 pigeons. � Prof. Tesler Ch. 1. Pigeonhole Principle Math 184A / Winter 2019 10 / 19
Proof of Generalized Pigeonhole Principle Put n items into k boxes (with n � 0 and k � 1 ). Then There is a box with at least ⌈ n / k ⌉ items. 1 There is a box with at most ⌊ n / k ⌋ items. 2 Let a i be the number of items in box i , for i = 1 , . . . , k . Each a i is a nonnegative integer, and a 1 + · · · + a k = n . Assume (by way of contradiction) that all boxes have fewer than ⌈ n / k ⌉ items: a i < ⌈ n / k ⌉ for all i . a i � ⌈ n / k ⌉ − 1 for all i . Thus, Then the total number of items placed in the boxes is a 1 + · · · + a k � k · ( ⌈ n / k ⌉ − 1 ) . � ������������� �� ������������� � = n Prof. Tesler Ch. 1. Pigeonhole Principle Math 184A / Winter 2019 11 / 19
Proof of Generalized Pigeonhole Principle 1. Show there is a box with at least ⌈ n / k ⌉ items Assume that every box has fewer than ⌈ n / k ⌉ items. Then the total number of items placed in the boxes is ( ∗ ) a 1 + · · · + a k � k · ( ⌈ n / k ⌉ − 1 ) � ������������� �� ������������� � = n Dividing n by k gives n = kq + r , where the quotient, q , is an integer and the remainder, r , is in the range 0 � r � k − 1 . If there is no remainder ( r = 0 ), then ⌈ n / k ⌉ = n / k = q . Inequality ( ∗ ) becomes n � k ( q − 1 ) = kq − k = n − k , so n � n − k . But n − k is smaller than n , a contradiction. If there is a remainder ( 1 � r � k − 1 ), then ⌈ n / k ⌉ = q + 1 . Inequality ( ∗ ) becomes n � k (( q + 1 ) − 1 ) = kq = n − r , so n � n − r . But n − r is smaller than n , a contradiction. Thus, our assumption that all a i < ⌈ n / k ⌉ was incorrect, so some box must have a i � ⌈ n / k ⌉ . Prof. Tesler Ch. 1. Pigeonhole Principle Math 184A / Winter 2019 12 / 19
Proof of Generalized Pigeonhole Principle 2. Show there is a box with at most ⌊ n / k ⌋ items Assume that every box has more than ⌊ n / k ⌋ items. Then the total number of items placed in the boxes is � k · ( ⌊ n / k ⌋ + 1 ) ( ∗ ) a 1 + · · · + a k � ������������� �� ������������� � = n Dividing n by k gives n = kq + r , where the quotient, q , is an integer and the remainder, r , is in the range 0 � r � k − 1 . ⌊ n / k ⌋ = q , so inequality ( ∗ ) becomes n � k ( q + 1 ) = kq + k = n − r + k , so n � n + ( k − r ) . But k − r > 0 , so n is smaller than n + ( k − r ) , a contradiction. Thus, our assumption that all a i > ⌊ n / k ⌋ was incorrect, so some box must have a i � ⌊ n / k ⌋ . Prof. Tesler Ch. 1. Pigeonhole Principle Math 184A / Winter 2019 13 / 19
Example: Ten points in the unit square 2D generalization of the 5 points in [ 0 , 1 ] problem (0,1) (1,1) (0,0) (1,0) Pick 10 points ( x 1 , y 1 ) , . . . , ( x 10 , y 10 ) in [ 0 , 1 ] × [ 0 , 1 ] . We’ll show that two of them have to be “close” together, within a certain small distance. Split [ 0 , 1 ] × [ 0 , 1 ] into a 3 × 3 grid of 1 / 3 × 1 / 3 squares. 10 points placed into 9 squares, and 10 > 9 , so some square has at least two points. Prof. Tesler Ch. 1. Pigeonhole Principle Math 184A / Winter 2019 14 / 19
Example: Ten points in the unit square (0,1) (1,1) (0,0) (1,0) The distance between two points in the same 1 / 3 × 1 / 3 square is √ 2 � � ( ∆ x ) 2 + ( ∆ y ) 2 � ( 1 / 3 ) 2 + ( 1 / 3 ) 2 = distance = 3 √ So the distance is at most 2 / 3 . √ 2 / 3 is achieved for opposite corners of a square (the red diagonals show some examples). Prof. Tesler Ch. 1. Pigeonhole Principle Math 184A / Winter 2019 15 / 19
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